Asymmetric radio frequency coils for magnetic resonance

ABSTRACT

Asymmetric radio frequency (RF) coils for magnetic resonance applications are provided. Also provided are time harmonic methods for designing such coils as well as symmetric coils. In addition, methods for converting complex current density functions into discrete capacitive and inductive elements are provided.

FIELD OF THE INVENTION

[0001] This invention relates to radio frequency coils for magneticresonance (MR) applications. In particular, the invention is directed toasymmetric radio frequency coils for magnetic resonance imaging (MRI)machines.

[0002] In certain of its aspects, the invention provides methods fordesigning radio frequency coils for magnetic resonance applicationswhich may be symmetric or asymmetric.

[0003] The radio frequency coils of the invention may be used fortransmitting a radio frequency field, receiving a magnetic resonancesignal, or both transmitting a radio frequency field and receiving amagnetic resonance signal. When the radio frequency coil serves atransmitting function, it will normally be combined with a shield toreduce magnetic interference with external components of the magneticresonance imaging system.

BACKGROUND OF THE INVENTION

[0004] In magnetic resonance imaging (MRI) applications, a patient isplaced in a strong and homogeneous static magnetic field, causing theotherwise randomly oriented magnetic moments of the protons, in watermolecules within the body, to precess around the direction of theapplied field. The part of the body in the homogeneous region of themagnet is then irradiated with radio-frequency (RF) energy, causing someof the protons to change their spin orientation. The net magnetizationof the spin ensemble is nutated away from the direction of the appliedstatic magnetic field by the applied RF energy. The component of thisnet magnetization orthogonal to the direction of the applied staticmagnetic field acts to induce measurable signal in a receiver coil tunedto the frequency of precession. This is the magnetic resonance (MR)signal.

[0005] The useful RF components are those generated in at plane at 90degrees to the direction of the static magnetic field. The same coilstructure that generates the RF field can be used to receive the MRsignal or a separate receiver coil placed close to the patient may beused. In either case the coils are tuned to the Larmor precessionalfrequency ω₀ where ω₀=γB₀ and γ is the gyromagnetic ratio for a specificnuclide and B₀ is the applied static magnetic field.

[0006] A desirable property of radio frequency coils for use in MR isthe generation of homogeneous RF fields over a prescribed region.Normally this region is central to the coil structure for transmissionresonators. A well known example of transmission resonators is thebirdcage resonator, details of which are given by Hayes et. al. in TheJournal of Magnetic Resonance, 63, 622 (1985) and U.S. Pat. No.4,694,255.

[0007] In some circumstances it is desirable to generate a target fieldover an asymmetric region of the coil structure, i.e., a region that isasymmetric relative to the mid-length point of the longitudinal axis ofthe coil structure. This is potentially advantageous for patient access,conformation of the coil structure to the local anatomy of the patientand for use in asymmetric magnet systems.

[0008] One method that is known in the art for generating homogeneousfields over a volume that is asymmetric to the coil structure is toenclose one end of the cylindrical structure, a so-called ‘end-cap’ ordome structure (details of which are given by Meyer and Ballon in TheJournal of Magnetic Resonance, 107, 19 (1995) and by Hayes in SMRM5^(th) annual meeting, Montreal, Book of Abstracts, 39 (1986)). Thesedesigns were applied to structures that surrounded only the head of apatient and, by their nature, prevent access to the top of the head. Thelimited access also makes these structures problematic for whole-bodyimaging as they substantially reduce access from one end of the magnet.

[0009] It is an object of this invention to provide coil structures thatgenerate desired RF fields within certain specific, and asymmetricportions of the overall coil structure, preferably without substantiallylimiting access from one end of the structure. Asymmetric radiofrequency coils can be used in conventional MR systems or in the newlydeveloped asymmetric magnets of U.S. Pat. No. 6,140,900.

[0010] It is a particular object of the present invention to provide ageneral systematic method for producing a desired radio frequency fieldwithin a coil, using a full-wave, frequency specific technique to firstdefine a current density on at least one cylindrical surface andsubsequently to synthesize a coil pattern from the current density.

[0011] It is a further particular object of the present invention to usecomplex current densities in the full-wave, frequency specific method.

SUMMARY OF THE INVENTION

[0012] In one broad form, the invention in accordance with certain ofits aspects provides a coil structure for a magnetic resonance devicehaving a cylindrical space with open ends, the coil structure beingadapted to generate a desired RF field within a specified portion of thecylindrical space. In accordance with the product aspects of theinvention, this portion is asymmetrically located relative to themid-length point of the longitudinal axis of the cylindrical space.

[0013] In connection with another aspect, the invention provides amethod for manufacturing a radio frequency coil structure for a MRdevice having a cylindrical space, preferably with open ends, comprisingthe steps of

[0014] selecting a target region over which a transverse RF magneticfield of a predetermined frequency is to be applied by the coilstructure, the target region being preferably asymmetrically locatedrelative to the mid-length point of the longitudinal axis of thecylindrical space,

[0015] calculating current density at the surface of the cylindricalspace required to generate the target field at the predeterminedfrequency,

[0016] synthesizing a design for the coil structure from the calculatedcurrent density in accordance with one of the methods discussed below,and

[0017] forming a coil structure according to the synthesized design.

[0018] Preferably, the method for calculating the current density uses atime harmonic method that accounts for the frequency of operation of theRF coil structure and makes use of a complex current density.

[0019] The RF coils of the invention can be used as transmitter coils,receiver coils, or both transmitter and receiver coils. As discussedabove, when the coil serves a transmitting function, it will normally becombined with a shield to reduce magnetic interference with externalcomponents of the magnetic resonance imaging system. To avoidredundancy, the following summary of the method aspects of the inventionis in terms of a RF coil system which includes a main coil(corresponding to the “first complex current density”) and a shieldingcoil (corresponding to the “second complex current density”), it beingunderstood that these methods can be practiced with just a main coil.

[0020] In accordance with a first method aspect of the invention, whichcan be used under “mild” coil length to wavelength conditions, i.e.,conditions in which the coil length is less than about one-fifth of theoperating wavelength, a method for designing apparatus for transmittinga radio frequency field (e.g., a field having a frequency of at least 20Megahertz, preferably at least 80 Megahertz), receiving a magneticresonance signal, or both transmitting a radio frequency field andreceiving a magnetic resonance signal is provided which comprises:

[0021] (a) defining a target region (e.g., a spherical or ellipsoidalregion preferably having a volume of at least 30×10³ cm³ andasymmetrically located relative to the midpoint of the RF coil, e.g.,the ratio of the distance between the midpoint of the target region andone end of the coil to the length of the coil is less than or equal to0.4) in which the radial magnetic component (e.g., B_(x), B_(y), orcombinations of B_(x) and B_(y)) of the radio frequency field is to havedesired values (e.g., substantially uniform magnitudes), said targetregion surrounding a longitudinal axis, i.e., the common longitudinalaxis of the magnetic resonance imaging system and the RF coil;

[0022] (b) specifying desired values for said radial magnetic componentof the radio frequency field at a preselected set of points within thetarget region (e.g., a set of points distributed throughout the targetregion or a set of points distributed at the outer boundary of thetarget region);

[0023] (c) defining a target surface (e.g., the shield coil surface)external to the apparatus on which the magnetic component of the radiofrequency field is to have a desired value of zero at a preselected setof points on said target surface;

[0024] (d) determining a first complex current density function, havingreal and imaginary parts, on a first specified cylindrical surface(i.e., the main coil surface) and a second complex current density,having real and imaginary parts, on a second specified cylindricalsurface (i.e., the shield coil surface), the radius of the secondspecified cylindrical surface being greater than the radius of the firstspecified cylindrical surface (e.g., the radius of the second surfacecan be 10% greater than the radius of the first surface) by:

[0025] (i) defining each of the complex current density functions as asum of a series of basis functions (e.g., triangular and/or pulse orsinusoidal and/or cosinusoidal functions) multiplied by complexamplitude coefficients having real and imaginary parts; and

[0026] (ii) determining values for the complex amplitude coefficientsusing an iterative minimization technique (e.g., a linear steepestdescent or a conjugate gradient descent technique) applied to a firstresidue vector obtained by taking the difference between calculatedfield values obtained using the complex amplitude coefficients at theset of preselected points in the target region and the desired values atthose points and a second residue vector equal to calculated fieldvalues obtained using the complex amplitude coefficients at thepreselected set of points on the target surface; and

[0027] (e) converting said first and second complex current densityfunctions into sets of capacitive elements (as understood by personsskilled in the art, such capacitive elements will in general have someinductive and resistive properties) and sets of inductive elements (asunderstood by persons skilled in the art, such inductive elements willin general have some capacitive and resistive properties) located on thespecified cylindrical surfaces by:

[0028] (i) converting each of the first and second complex currentdensity functions into a curl-free component J_(curl-free) and adivergence-free component J_(div-free) using the relationships:

J_(curl-free)=∇ψ, and

J _(div-free) =∇×S,

[0029] where ψ and S are functions obtained from the respective firstand second complex current density functions through the equations:

∇² ψ=∇·J,

−∇² S=∇×J, and

−∇²(n·S)=n·∇×J,

[0030] where n is a vector normal to the respective first and secondspecified cylindrical surfaces and J is the respective first and secondcomplex current density functions;

[0031] (ii) calculating locations on the respective first and secondcylindrical surfaces for the respective sets of capacitive elements bycontouring the respective ψ functions; and

[0032] (iii) calculating locations on the respective first and secondcylindrical surfaces for the respective sets of inductive elements bycontouring the respective functions n·S (ψ and n·S are referred toherein and function as “streaming functions”).

[0033] In accordance with a second method aspect of the invention, whichcan be used under “non-mild” coil length to wavelength conditions, i.e.,conditions in which the coil length can be greater than about one-fifthof the operating wavelength, a method for designing apparatus fortransmitting a radio frequency field or both transmitting a radiofrequency field and receiving a magnetic resonance signal is providedwhich comprises:

[0034] (a) defining a target region in which the radial magneticcomponent of the radio frequency field is to have desired values, saidtarget region surrounding a longitudinal axis, i.e., the commonlongitudinal axis of the magnetic resonance imaging system and the RFcoil;

[0035] (b) specifying desired values for said radial magnetic componentof the radio frequency field at a preselected set of points within thetarget region;

[0036] (c) defining a target surface external to the apparatus on whichthe magnetic component of the radio frequency field is to have a desiredvalue of zero;

[0037] (d) determining a first complex current density function, havingreal and imaginary parts, on a first specified cylindrical surface and asecond complex current density, having real and imaginary parts, on asecond specified cylindrical surface, the radius of the second specifiedcylindrical surface being greater than the radius of the first specifiedcylindrical surface:

[0038] (i) defining each of the complex current density functions as asum of a series of basis functions (e.g., sinusoidal and/or cosinusoidalfunctions) multiplied by complex amplitude coefficients having real andimaginary parts; and

[0039] (ii) determining values for the complex amplitude coefficients bysimultaneously solving matrix equations of the form:

[A ₁ ^(C)](a ^(C))+[A ₁ ^(S)](a ^(S))=B ^(C)   (Eq. I)

[A ₂ ^(C)](a ^(C))+[A ₂ ^(S)](a ^(S))=B ^(S)

[0040] where A₁ ^(C), A₁ ^(S), A₂ ^(C), and A₂ ^(S) are transformationmatrices between current density space and magnetic field space whosecomponents are based on time harmonic Green's functions, a^(C) and a^(S)are vectors of the unknown complex amplitude coefficients for the firstand second complex current density functions, respectively, B^(C) is avector of the desired values for the radial magnetic field specified instep (b), and B^(S) is a vector whose values are zero, said equationsbeing solved by:

[0041] (1) transforming the equations into functionals that can besolved using a preselected regularization technique, and

[0042] (2) solving the functionals using said regularization techniqueto obtain values for the complex amplitude coefficients; and

[0043] (e) converting said first and second complex current densityfunctions into sets of capacitive elements and sets of inductiveelements located on the specified cylindrical surfaces (preferably, thisstep is performed using the methods of step (e) of the first methodaspect of the invention; also the methods of said step (e) can be usedindependent of either the first or second method aspects of theinvention to convert a complex current density function to sets ofcapacitive and inductive elements, i.e., to a manufacturable coilstructure).

[0044] In accordance with certain preferred embodiments of this secondmethod aspect of the invention, the regularization functional is chosenso as to minimize the integral of the dot product of the first complexcurrent density function with itself over the first specifiedcylindrical surface and to minimize the integral of the dot product ofthe second complex current density function with itself over the secondspecified cylindrical surface. Other regularization functionals that canbe used include: (1) the second derivative of the complex currentdensity functions, and (2) other functionals besides the dot productthat are proportional to the square of the complex current densityfunctions.

[0045] In accordance with other preferred embodiments, the complexamplitude coefficients are chosen so that the first and second complexcurrent density functions each has zero divergence.

[0046] The parenthetical statements set forth in connection with thesummary of the first method aspect of the invention also apply to thesecond method aspect of the invention except where indicated. Inconnection with all of its method aspects, the invention also preferablyincludes the additional step(s) of displaying the locations of the setsof inductive elements on the first and second specified cylindricalsurfaces and/or producing physical embodiments of those sets ofelements.

[0047] In accordance with certain of its product aspects, the inventionprovides apparatus for use in a magnetic resonance system fortransmitting a radio frequency field (e.g., a field having a frequencygreater than, for example, 20 megahertz, and preferably greater than 80megahertz), receiving a magnetic resonance signal, or transmitting aradio frequency field and receiving a magnetic resonance signal, saidapparatus and said magnetic resonance imaging system having a commonlongitudinal axis, said apparatus comprising:

[0048] (a) a support member (e.g., a tube composed of fiberglass,TEFLON, or other materials, which preferably can be used as a substratefor a photolithography procedure for printing conductive patterns andwhich does not substantially absorb RF energy) which defines a bore(preferably a cylindrical bore) having first and second open ends whichare spaced from one another along the longitudinal axis by a distance L(the open ends are preferably both sized to receive a patient's bodypart which is to be imaged, e.g., the head, the upper torso, the lowertorso, a limb, etc.); and

[0049] (b) a plurality of inductive elements (e.g., copper or othermetallic tracks or tubes) and a plurality of capacitive elements (e.g.,distributed or lumped elements) associated with the support member(e.g., mounted on and/or mounted in and/or mounted to the supportmember);

[0050] wherein if used as a transmitter, the apparatus has the followingcharacteristics (the “if used as a transmitter” terminology is used indefining both transmitting and receiving RF coils since by reciprocity,coils that transmit uniform radio frequency fields, receive radiofrequency fields with a uniform weighting function):

[0051] (i) the apparatus produces a radio frequency field which has aradial magnetic component which has a spatially-varying peak magnitudewhose average value is A_(r-avg);

[0052] (ii) the apparatus has a homogenous volume within the bore overwhich the spatially-varying peak magnitude of the radio frequency fieldhas a maximum deviation from A_(r-avg) which is less than or equal to15% (preferably less than or equal to 10%);

[0053] (iii) the homogeneous volume defines a midpoint M which is on thelongitudinal axis, is closer to the first end than to the second end,and is spaced from the first end by a distance D such that the ratio D/Lis less than or equal to 0.4 (preferably less than or equal to 0.25);and

[0054] (iv) at least one of said inductive elements (e.g., 1, 2, 3, . .. or all) comprises a discrete conductor (e.g., a copper or othermetallic track or tube) which follows a sinuous path such that duringuse of the apparatus current flows through a first part of the conductorin a first direction that has both longitudinal and azimuthal componentsand through a second part of the conductor in a second direction thathas both longitudinal and azimuthal components, said first and seconddirections being different. An example of such a sinuous path is shownin FIG. 15 discussed below.

[0055] In certain preferred embodiments, the homogenous volume is atleast 30×10³ cm³ and/or L is at least 1.0 meter.

[0056] In other preferred embodiments, the first and second open endshave transverse cross-sectional areas A₁ and A₂, respectively, whichsatisfy the relationship:

|A ₁ −A ₂ |/A ₁<0.1.

[0057] where A₁ is preferably at least 2×10³ cm².

[0058] In connection with all of the aspects of the invention, thelengths of coils and of cylindrical spaces associated therewith aredefined in terms of the inductive elements making up the coil. Inparticular, for a horizontally oriented coil, the length constitutes thedistance along the longitudinal axis from the leftmost edge of theleftmost inductive element to the rightmost edge of the rightmostinductive element, corresponding definitions applying to otherorientations of the coil.

BRIEF DESCRIPTION OF THE DRAWINGS

[0059] The invention will be described by way of examples with referenceto the drawings in which:

[0060]FIG. 1 illustrates a general layout of a cylindrical radiofrequency coil system of length L and radius a. Changing the value of Dchanges the degree of symmetry of the ‘target region’ relative to thecoil structure;

[0061]FIG. 2 shows a current distribution on the surface of a cylindercalculated using a quasi-static formulation, where the vertical axis isthe z-axis;

[0062]FIG. 3 shows plots of J_(φ) and J_(z) along the length of acylinder in the φ=0 and φ=π/2 planes respectively for a quasi-staticformulation;

[0063]FIGS. 4A and 4B show plots of J_(φ) and J_(z) along the length ofa cylinder in the φ=0 and φ=π/2 planes respectively for the realsolutions of Table 4 (190 MHz). In FIG. 4A, z_(os)=20 mm, while in FIG.4B, z_(os)=50 mm;

[0064]FIG. 5 shows the current distribution on the surface of a cylindercalculated from the real solution of Table 4 (z_(os)=20 mm; 190 MHz);

[0065]FIG. 6 shows the current distribution on the surface of a cylindercalculated from the real solution of Table 4 (z_(os)=50 mm; 190 MHz);

[0066]FIG. 7 shows a full conductor model of the contours of thesymmetric current density distribution shown in FIG. 3.

[0067]FIG. 8 shows a current density distribution for a 320 mm lengthcoil with radius 100 mm having a homogeneous volume 50 mm from thecenter of the coil towards the coil's lower end.

[0068]FIG. 9 shows contour lines for n·S for the current density of FIG.8.

[0069]FIG. 10 shows contour lines for ψ for the current density of FIG.8.

[0070]FIG. 11 shows the contour lines for the stream function n·Scalculated using the second method aspect of the invention.

[0071]FIG. 12 shows a normalized magnetic field for a representativeasymmetric coil in the transverse plane determined using the secondmethod aspect of the invention. The upper curve shows the distributionin the x direction and the lower curve shows the distribution in the ydirection.

[0072]FIG. 13 shows a transverse magnetic field along the z-axis for arepresentative asymmetric coil determined using the second method aspectof the invention.

[0073]FIG. 14 is a circuit-type layout for a representative asymmetriccoil.

[0074]FIG. 15 is a 3D perspective of an inductive, geometry for arepresentative asymmetric coil. A partial shield is included forillustrative purposes.

[0075]FIG. 16 is a flow chart illustrating the first method aspect ofthe invention.

[0076]FIG. 17 is a flow chart illustrating the second method aspect ofthe invention.

[0077]FIG. 18 is a flow chart illustrating a preferred method forcalculating a coil geometry from a current density.

[0078] The foregoing drawings, which are incorporated in and constitutepart of the specification, illustrate the preferred embodiments of theinvention, and together with the description, serve to explain theprinciples of the invention. It is to be understood, of course, thatboth the drawings and the description are explanatory only and are notrestrictive of the invention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

[0079] As discussed above, the present invention relates to RF coilshaving prescribed properties and to methods for designing these andother types of RF coils. FIGS. 16-18 illustrate the overall numericalprocedures of the invention with reference to the various equationspresented below.

[0080] The methods of the invention as described below are preferablypracticed on a digital computer system configured by suitableprogramming to perform the various computational steps. The programmingcan be done in various programming languages known in the art. Apreferred programming language is the C language which is particularlywell-suited to performing scientific calculations. Other languages whichcan be used include FORTRAN, BASIC, PASCAL, C⁺⁺, and the like. Theprogram can be embodied as an article of manufacture comprising acomputer usable medium, such as a magnetic disc, an optical disc, or thelike, upon which the program is encoded.

[0081] The computer system can comprise a general purpose scientificcomputer and its associated peripherals, such as the computers andperipherals currently being manufactured by DIGITAL EQUIPMENTCORPORATION, IBM, HEWLETT-PACKARD, SUN MICROSYSTEMS, SGI or the like.For example, the numerical procedures of the invention can beimplemented in C-code and performed on a work station. The system shouldinclude means for inputting data and means for outputting the results ofthe RF coil design both in electronic and visual form. The output canalso be stored on a disk drive, tape drive, or the like for furtheranalysis and/or subsequent display.

[0082] Quasi-Static Analysis

[0083] In outline, the object of the design process is to producestructures that generate target RF transverse fields (that is, B_(x) orB_(y) or combinations thereof) over a defined volume inside a cylinder.The value of B_(x) (for example) is specified at a number of chosenpoints inside this region and in one embodiment, inverse methods areused to compute the corresponding current distribution.

[0084] The geometry of the coil is shown in FIG. 1. Surface currentdensities on the cylinder are defined as:

J(r)=J _(φ)(φ,z){circumflex over (φ)}+J _(z)(φ,z){circumflex over (z)}for ρ=a and −L/2≦z≦L/2   (1)

[0085] where r=ρ{circumflex over (ρ)}+z{circumflex over (z)}; L and aare the length and radius of the cylinder.

[0086] As is known in the art, in one formulation for the case of asymmetric coil structure, J(r) can be written as a summation of Msinusoidal basis functions: $\begin{matrix}{{J_{\varphi}( {\varphi,z} )} = {\cos \quad \varphi {\sum\limits_{m = 1}^{M}\quad {J_{c\quad m}{\sin ( {k_{m}z} )}}}}} & (2) \\{{J_{z}( {\varphi,z} )} = {\sin \quad \varphi {\sum\limits_{n = 1}^{M}\quad {\frac{- J_{c\quad m}}{k_{m}a}{\cos ( {k_{m}z} )}}}}} & (3)\end{matrix}$

[0087] where J_(cm) are the current coefficients and k_(m)=(2m−1)π/L.

[0088] However, this expression is only applicable to problems where themagnetic field is symmetrical about z=0. Magnetic fields are expressedin terms of magnetic vector potentials defined as the convolution ofsurface current density with the corresponding Green's function in thespatial domain: $\begin{matrix}{{A(r)} = {{\frac{\mu_{0}a}{4\pi}{\int_{{- L}/2}^{L/2}{\int_{0}^{2\pi}{{G( r \middle| r^{\prime} )}{J( r^{\prime} )}\quad {\varphi^{\prime}}\quad {z^{\prime}}\quad {for}\quad \rho}}}} = a}} & (4)\end{matrix}$

[0089] where the dashed elements indicate source points and the undashedthe field points. The corresponding generalized Green's function inCartesian coordinates is: $\begin{matrix}{{G( r \middle| r^{\prime} )} = \frac{^{{- j}\quad k_{0}{{r - r^{\prime}}}}}{{r - r^{\prime}}}} & (5)\end{matrix}$

[0090] where r and r′ are the position vectors of field and sourcepoints respectively; k₀ is the wave-number given by k₀=2π/λ₀; and λ₀ isthe free-space wavelength of the propagating field.

[0091] We consider first an initial quasi-static simplification in whichthe wavelength is assumed to be very long, or the frequency near zero.Equation (5) then approaches: $\begin{matrix}{{G( r \middle| r^{\prime} )} = \frac{1}{{r - r^{\prime}}}} & (6)\end{matrix}$

[0092] In cylindrical coordinates one solution of this is:$\begin{matrix}{{G( r \middle| r^{\prime} )} = {\frac{2}{\pi}{\sum\limits_{p = {- \infty}}^{+ \infty}\quad {^{j\quad {p{({\varphi - \varphi^{\prime}})}}}{\int_{0}^{\infty}\quad {{k}\quad {\cos \lbrack {k( {z - z^{\prime}} )} \rbrack}{I_{p}( {k\quad \rho} )}{K_{p}( {k\quad \rho^{\prime}} )}}}}}}} & (7)\end{matrix}$

[0093] where r(r′) are again defined as the position vectors of field(source) respectively. I_(p) and K_(p) are the modified Bessel functionsof p^(th) order. This solution is for the case where observation pointsare internal to the cylinder, i.e., ρ′>ρ.

[0094] The vector potentials may then be expressed as: $\begin{matrix}{{A_{\rho}(r)} = {\frac{\mu_{0}a}{4\pi}{\int_{{- L}/2}^{L/2}{\int_{0}^{2\pi}{{G( r \middle| r^{\prime} )}{J_{\varphi}( r^{\prime} )}\quad {\sin ( {\varphi - \varphi^{\prime}} )}{\varphi^{\prime}}\quad {z^{\prime}}}}}}} & (8) \\{{A_{\varphi}(r)} = {\frac{\mu_{0}a}{4\pi}{\int_{{- L}/2}^{L/2}{\int_{0}^{2\pi}{{G( r \middle| r^{\prime} )}{J_{\varphi}( r^{\prime} )}{\cos ( {\varphi - \varphi^{\prime}} )}\quad {\varphi^{\prime}}\quad {z^{\prime}}}}}}} & (9) \\{{A_{z}(r)} = {\frac{\mu_{0}a}{4\pi}{\int_{{- L}/2}^{L/2}{\int_{0}^{2\pi}{{G( r \middle| r^{\prime} )}{J_{z}( r^{\prime} )}{\varphi^{\prime}}\quad {z^{\prime}}}}}}} & (10)\end{matrix}$

[0095] Substituting equation (7) into the above equations and carryingout the integration over the cylinder's surface, we obtain theexpression for (8)-(10) in the form of Fourier integral (series) or interms of spectral variable k (m). The relevant magnetic fields arecalculated from the curl of the vector potentials to be: $\begin{matrix}{{B_{\rho}(r)} = {\frac{{j\mu}_{0}a}{\pi}{\sum\limits_{p = {- \infty}}^{\infty}{^{j\quad p\quad \varphi}{\int_{0}^{\infty}{k\quad \cos \quad {kz}\quad {J_{\varphi}( {m,k} )}{I_{p}^{\prime}\quad( {k\quad \rho} )}{K_{p}^{\prime}({ka})}{k}}}}}}} & (11) \\{{B_{\varphi}(r)} = {{- \frac{{j\mu}_{0}a}{\pi \quad \rho}}{\sum\limits_{p = {- \infty}}^{\infty}{p\quad ^{j\quad p\quad \varphi}{\int_{0}^{\infty}{\cos \quad {kz}\quad {J_{\varphi}( {m,k} )}{I_{p}\quad( {k\quad \rho} )}{K_{p}^{\prime}({ka})}{k}}}}}}} & (12)\end{matrix}$

[0096] Writing the currents as summation of (2) and (3) and using thefollowing relationship:

B _(x) =B _(ρ) cos φ−B _(φ) sin φ  (13)

[0097] we obtain $\begin{matrix}{{{B_{x}(r)} = {{- \frac{\mu_{0}{aL}}{2\pi \quad \rho}}{\int_{0}^{\infty}{\cos \quad {kz}\quad {\sum\limits_{m = 1}^{M}\quad {J_{c\quad m}{\Psi_{m}(k)}{{K_{1}^{\prime}({ka})}\lbrack {{k\quad \rho \quad \cos^{2}\varphi \quad {I_{0}( {k\quad \rho} )}} - {\cos \quad 2\varphi \quad {I_{1}( {k\quad \rho} )}}} \rbrack}{k}}}}}}}\quad} & (14) \\{where} & \quad \\{{\Psi_{m}(k)} = {\frac{{\sin ( {k - k_{m}} )}{L/2}}{( {k - k_{m}} ){L/2}} - \frac{{\sin ( {k + k_{m}} )}{L/2}}{( {k + k_{m}} ){L/2}}}} & (15)\end{matrix}$

[0098] The result described in equations (14) and (15) is known in theart as a quasi-static solution, being only valid in applications wherethe free space wavelength is much longer than the length of the cylindergenerating the RF fields (see for example, H. Fujita, L. S. Petropoulos,M. A. Morich, S. M. Shvartsman, and R. W. Brown ‘A hybrid inverseapproach applied to the design of lumped-element RF coils,’ IEEE Trans.Biomedical Engineering, vol-46, pp. 353-361, March 1999).

[0099] A target region is then defined as the volume within which ahomogenous B_(x) is required. In particular, a target field value(B_(xt)) is defined as the fixed magnitude of the desired homogenousfield B_(x). The target region is, for example, specified as the volumeinside a sphere of radius R centered at z=0. As an example, a targetspecification could be that R is set to be R≈0.7a andmax(B_(x))−min(B_(x)) is specified at less than 5%. If we setB_(x)(r₁)=B_(x)(r₂)=B_(x)(r₃)= . . . =B _(x)(r_(N))=B_(xt) where r₁, r₂,r₃ . . . r_(N) are the target field points chosen inside the targetregion, a set of N linear equations is obtained from equation (14).

[0100] In particular, the following matrix equation is obtained:

B _(xt)×[1]_(N×1) =[T] _(N×M) [J] _(M×1)   (16)

[0101] where [T] is a matrix containing the values of B_(x) at thedefined observation points due to each of the current components;[J_(cm)] is a column vector of the current coefficients J_(c1), J_(c2),J_(c3) . . . J_(cm) and [1] is a unity column vector.

[0102] If the number of linear equations equals the number of unknowns(M=N), then the numerical solution for [J_(cm)] can be easily obtainedby multiplying both sides of (16) by [T]⁻¹.

[0103] As an example of the quasi-static method, a coil of dimension:a=100 mm, L=320 mm was designed. The specified target field in sphericalcoordinates was: B_(xt)= B_(x) r₁ r₂ r₃ φ₁ φ₂ φ₃ θ₁ θ₂ θ₃ 23.5 μT 50 5050 (mm) 90° 90° 45° 90° 0 45°

[0104] The current density coefficients up to M=3 were then calculatedfrom the quasi-static expressions in equations (14) to be: J_(c1) J_(c2)J_(c3) 43.834274291992190 −48.744728088378910 36.222351074218750

[0105] The corresponding resultant current distribution on the surfaceof the cylinder is shown in FIG. 2 and illustrates the current pathsrequired for the coil. In the graph of FIG. 3, the current componentsJ_(φ) and J_(z) are plotted along the z-direction for φ=0 and φ=α/2respectively. Contour plots (not shown) demonstrate that B_(x) ishomogeneous within the target region.

[0106] Time Harmonic Analysis

[0107] In accordance with the invention, instead of using thequasi-static approach, a time harmonic, full wave method is used tocalculate the required current density. This is essential in cases wherethe operating wavelength and the dimension of the cylinder are of thesame order, as the prior art quasi-static method described in the abovesection is no longer valid. However, the mathematical derivation of atime-harmonic expression for B_(x) in the spectral domain or anequivalent of (14) is extremely difficult. A more direct approach is todefine B_(x) directly in terms of the vector potentials of thegeneralized Green's function of equation (5) rather than its spectralequivalent given in equation (7): $\begin{matrix}{B_{x} = {\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}}} & (17)\end{matrix}$

[0108] where A_(z) are A_(y) are the vector potentials calculated fromequation (4). (Note that numerically, the calculation of equation (17)is carried out using a Gaussian-Quadrature integration routine over theprescribed 2-D cylindrical surface integral.)

[0109] In the time-harmonic case, the wavelength or frequency ofoperation is taken into account and the wavenumber (k₀) is non-zero.This leads in many instances to the current density and its constituentcoefficients being complex (that is, consisting of both real andimaginary components). In accordance with the invention, the calculatedcurrent densities for the generation of specific transverse RF fieldsare such complex quantities.

[0110] To illustrate the use of complex current densities, currentdensities were calculated on the same cylinder as the quasi-staticexample discussed above at three different frequencies: 190 MHz, 300 MHzand 500 MHz. The same target field was used, i.e., the homogeneousvolume was symmetrically located. Equation (16) was modified to includecomplex current densities and a complex version of the matrix [T] wasobtained using equation (17). The solution for [J] then follows the sameprocedures for the quasi-static example except that complex quantitiesare now included. The calculated current coefficients are shown in thefollowing table: TABLE 1 Computed current coefficients (M = 3) atdifferent frequencies. Frequency (MHz) J_(c1) J_(c2) J_(c3) 19038.6895 + j1.9310 −42.7225 − j2.1026  35.0854 + j1.6609 300 33.4992 +j5.4433 −36.6628 − j5.7392  35.1309 + j4.9677 500 23.8764 + j13.0168−25.3327 − j12.4308 39.07675 + j14.0778

[0111] From the above table, it is clear that as the frequencyincreases, the imaginary part of the current densities becomes larger.

[0112] Basis Functions for Asymmetric Systems

[0113] In a preferred embodiment of the invention, open-ended asymmetricstructures are designed for a time harmonic system. For the design ofasymmetric coil structures, surface current densities (Amp/meter) on acylinder are defined in the same manner as for symmetric structures:

J(r)=J _(φ)(φ,z){circumflex over (φ)}+J _(z)(φ,z){circumflex over (z)}for ρ=a and −L/2≦z≦L/2   (18)

[0114] The relationship between J_(z) and J_(φ), under mild coillength/wavelength ratios, may be approximated as: $\begin{matrix}{{J_{\varphi}( {\varphi,z} )} = {{- a}\quad {\int{\frac{J_{z}}{z}{\varphi}}}}} & (19)\end{matrix}$

[0115] Or alternately J_(z) and J_(φ) may be decoupled and treatedindividually for the purposes of optimization. Currents on the coil aredescretized along z and are represented by sub-domain basis functions.

[0116] For asymmetric structures a new set of basis functions must bespecified, and in one embodiment J_(z) and J_(φ) are written as asummation of triangular T_(z)( ) and pulse Π_(z)( ) functionsrespectively as follows: $\begin{matrix}{{J_{z}( {\varphi,z} )} = {\cos \quad \varphi \quad {\sum\limits_{m = 1}^{M}{J_{c\quad m}{T_{z}( {z - z_{m}} )}}}}} & (20) \\{{J_{\varphi}( {\varphi,z} )} = {\frac{a}{l_{z}}\quad \sin \quad \varphi \quad {\sum\limits_{n = 1}^{M}{{- J_{c\quad m}}\{ {{\Pi_{z}( {z - z_{m}^{+}} )} - {\Pi_{z}( {z - z_{m}^{-}} )}} \}}}}} & (21)\end{matrix}$

[0117] where the J_(cm) are the current coefficients and $\begin{matrix}{{T_{z}( {z - z_{m}} )} = \{ \begin{matrix}{1 - {{{z - z_{m}}}/l_{z}}} & {{{z - z_{m}}} < l_{z}} \\0 & {elsewhere}\end{matrix} } & (22) \\{{\Pi_{z}( {z - z_{m}^{\pm}} )} = \{ \begin{matrix}1 & {{{z - z_{m}^{\pm}}} < {l_{z}/2}} \\0 & {elsewhere}\end{matrix} } & (23) \\{{{{and}\quad z_{m}} = {{m\quad l_{z}} - \frac{L}{2}}},{z_{m}^{\pm} = {{{( {m \pm \frac{1}{2}} )l_{z}} - {\frac{L}{2}\quad {and}\quad l_{z}}} = {L/{( {M + 1} ).}}}}} & \quad\end{matrix}$

[0118] Magnetic fields are expressed in terms of magnetic vectorpotentials defined as the convolution of surface current density withthe corresponding Green's function in the spatial domain:$\begin{matrix}\begin{matrix}{{A(r)} = {\frac{\mu_{0}a}{4\quad \pi}{\int_{{- L}/2}^{L/2}{\int_{0}^{2\quad \pi}{{G( r \middle| r^{\prime} )}{J( r^{\prime} )}{\varphi^{\prime}}{z^{\prime}}}}}}} & {{{for}\quad \rho} = a}\end{matrix} & (24)\end{matrix}$

[0119] As previously, the corresponding generalized Green's function incartesian coordinates is: $\begin{matrix}{{G( r \middle| r^{\prime} )} = \frac{^{{- j}\quad k_{0}{{r - r^{\prime}}}}}{{r - r^{\prime}}}} & (25)\end{matrix}$

[0120] where r and r′ are once again the position vectors of field andsource points respectively; k₀ is the wave-number given by k₀=2π/λ₀; andλ₀ and is the free-space wavelength of the propagating field.

[0121] The transverse field is written as: $\begin{matrix}{B_{x} = {\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}}} & (26)\end{matrix}$

[0122] where A_(z) are A_(y) are the vector potentials calculated byequation (24).

[0123] In accordance with the invention, the homogeneous field ispreferably asymmetrical about the z=0 plane. The target region isdefined as a spherical volume of radius R centered at z=z_(os). Ourfirst method of obtaining a set of current coefficients that producesthe required homogeneity is as follows.

[0124] Let (r₁, r₂, r₃ . . . r_(N)) be the N field positions choseninside the target region and B_(x)(r_(n))(n=1 . . . N) be the magneticfield at these points. The field at each target point is expressed interms of M number of current coefficients as follows: $\begin{matrix}{{B_{x}( r_{n} )} = {\sum\limits_{m = 1}^{M}{C_{mn}J_{cm}}}} & (27)\end{matrix}$

[0125] where C_(mn) represents the field contribution at r_(n) due tothe basis function at z=z_(m).

[0126] The residue (R_(n)) is defined to be the difference betweenB_(x)(r_(n)) and the specified target field B_(xt), i.e.,

R _(n)(J ₁ ,J ₂ . . . J _(N))=B _(x)(r _(N))−B _(xt) n=1 . . . N   (28)

[0127] A function f is defined ƒ as: $\begin{matrix}{{f( {J_{1},{J_{2}\quad \ldots \quad J_{N}}} )} = {\sum\limits_{n = 1}^{N}{R_{n}^{2}( {J_{1},{J_{2}\quad \ldots \quad J_{N}}} )}}} & (29)\end{matrix}$

[0128] In order to suppress large current variations in the solution, itis sometimes useful to also include the change of current amplitude as aconstrained condition. In this case a term$w\quad {\sum\limits_{i = 1}^{N - 1}m_{i}}$

[0129] is added to equation (28), where m_(i) is the difference betweenthe adjacent current elements given by m_(i)=J_(i)−J_(i+1); w is theweight used for bringing m_(i) to the same order of magnitude as R_(n).

[0130] Our task is to find a set of {J₁,J₂ . . . J_(i+1)} such that ƒ isminimum, or ƒ_(min)=min{ƒ(J₁,J₂ . . . J_(M))}. There are various ways inwhich this function may be minimized. In one embodiment, a linearsteepest descent method is used for each variable. An advantage of usingthis minimization technique is that it allows either a real or a complexsolution to be readily obtained.

EXAMPLES OF THE FIRST METHOD ASPECT OF THE INVENTION

[0131] Examples of the invention, designed using the time harmonic, fullwave method of the first method aspect of the invention are nowdetailed.

[0132] A coil of dimension: a=100 mm, L=320 mm was designed at differentfrequencies: 0 MHz, 190 MHz and 300 MHz. A set of target points (r_(n))was defined in spherical coordinates with respect to z_(os), where interms of the parameters of FIG. 1, |z_(os)|+|D|=L/2. The radialpositions of these points were r_(n)=53.3 mm; their angular positionsare given in the following table: TABLE 2 Angular positions of targetpoints in spherical coordinates centered at z_(os). N φ θ 1  0°  0° 245°  0° 3 90°  0° 4 45° 30° 5  0° 45° 6 45° 45° 7 45° 60° 8  0° 90° 990° 90° 10 45° 120°  11 45° 135°  12 45° 150°  13  0° 180° 

[0133] The specified target field was B_(xt)=23.5 μT. The requirementfor homogeneity was specified as: max(B_(x))−min(B_(x))<5% inside aspherical region of radius R≈0.7a for z_(os)=20 mm and R≈0.55a forz_(os)=50 mm. Eight current coefficients (M) were used for eachcomputation. Tables 3 to 5 show the calculated current coefficients forthe different z_(os) values at the three operating frequencies. For thenon-static cases, the real (Re) and imaginary (Im) components of thecomplex currents are shown separately. These coefficients are for usewith the triangular and pulse functions discussed above. TABLE 3Computed current coefficients (M = 8) for quasi-static case. Z_(os)J_(c1) J_(c2) J_(c3) J_(c4) 20 mm 220.68 −188.62 −7.60 −42.38 50 mm1763.41 −959.25 −163.87 −52.156 Z_(os) J_(c5) J_(c6) J_(c7) J_(c8) 20 mm−29.62 −36.10 −25.10 −62.20 50 mm −18.58 −35.02 −9.71 −69.13

[0134] TABLE 4 Computed complex current coefficients (M = 8) at 190 MHz.Z_(os(mm)) Re/Im J_(c1) J_(c2) J_(c3) J_(c4) 20 Re 226.65 −195.19 17.93−42.11 Im −159.73 +71.66 −5.96 −4.30 50 Re 815.32 −280.51 −68.98 0.22 Im−505.05 167.50 5.94 −2.68 Z_(os(mm)) Re/Im J_(c5) J_(c6) J_(c7) J_(c8)20 Re −24.42 −33.19 −20.73 −55.24 Im +0.65 −1.77 −0.79 −1.81 50 Re−33.27 −30.49 −9.94 −63.67 Im 0.21 0.98 0.47 0.44

[0135] TABLE 5 Computed complex current coefficients (M = 8) at 300 MHz.Z_(os(mm)) Re/Im J_(c1) J_(c2) J_(c3) J_(c4) 20 Re 194.73 — −14.29−22.91 Im — 117.21 −21.10 −1.15 50 Re 634.66 −48.97 — 23.80 Im 0.92 3.099.78 4.87 Z_(os(mm)) Re/Im J_(c5) J_(c6) J_(c7) J_(c8) 20 Re −23.01−29.57 −10.18 −51.95 Im −1.32 −4.98 0.61 −10.29 50 Re −36.71 −27.10−8.53 −53.72 Im 4.19 3.67 2.50 2.32

[0136] The transverse magnetic field (B_(x)) inside the cylinder wasexamined for each of the above cases. Computed results show that theresultant B_(x) values meet the required homogeneity for all the abovecases and they show similar characteristics for a given z_(os). For the190 MHz case, the real components of the current densities J_(φ) andJ_(z) are plotted along the z-direction in the φ=0 and φ=π/2 planes onthe graphs of FIG. 4A and 4B which correspond to z_(os=)20 mm andz_(os)=50 mm respectively. In both cases, J_(φ) and J_(z) along z are ofan oscillatory nature. It is seen that both components experience asurge in magnitude near the positive end of the coil (i.e., the endclosest to the end where the target region is located).

[0137]FIGS. 5 and 6 show current distributions on the surface of acylinder from the real solutions of Table 4 for z_(os) equal to 20 mmand 50 mm, respectively.

[0138] Plots of the B_(x) field created by the currents of FIG. 4A andFIG. 4B over the volume ρ≦80 mm, −160 mm≦z≦160 mm (not shown) revealed ahomogeneous region in both cases that was asymmetrical about the centerof the coil and satisfied the criteria for homogeneity. It was alsofound that there was a larger volume of homogeneity in the z_(os)=20 mmcase than in the z_(os)=50 mm case.

EXAMPLES OF THE SECOND METHOD ASPECT OF THE INVENTION

[0139] As a representative illustration of the second method aspect ofthe invention, the following sets of basis functions are used for thecurrent density components on the surface of a cylinder with radius ρ₀(mn) and length L (m): $\begin{matrix}{J_{\varphi} = {\sum\limits_{q = 0}^{1}{\sum\limits_{p = 0}^{1}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = {p + 1}}^{M}{a_{mnpq}{\cos ( {{n\quad \varphi} + \frac{q\quad \pi}{2}} )}{\cos ( {{k_{m}z} + \frac{( {{2p} - m} )\pi}{2}} )}{\hat{a}}_{\varphi}}}}}}} & (30) \\{J_{z} = {\sum\limits_{q = 0}^{1}{\sum\limits_{p = 0}^{1}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = {p + 1}}^{M}{c_{mnpq}{\sin ( {{n\quad \varphi} + \frac{q\quad \pi}{2}} )}{\sin ( {{k_{m}z} + \frac{( {{2p} - m} )\pi}{2}} )}{\hat{a}}_{z}}}}}}} & (31) \\{where} & \quad \\{k_{m} = \frac{( {m - p} )\pi}{L}} & (32)\end{matrix}$

[0140] and the complex coefficients a_(mnpq) and c_(mnpq) are to becalculated.

[0141] This form of the basis functions was selected based on thefollowing rationale. If the specified field has only one vectorcomponent that does not vary with φ, then it can be reasoned that the$\frac{q\quad \pi}{2}$

[0142] term in the basis functions is unnecessary as the axis can be setin any direction in the xy plane. However for simplicity, φ=0 is set tocoincide with the x-axis and the $\frac{q\quad \pi}{2}$

[0143] term will be necessary for any specified field that has ay-component.

[0144] For specified fields in MRI applications, it is expected that thecurrent density J is anti-symmetrical at 180°, so coefficients of evenharmonics of φ will be zero and hence, n=1, 3, 5 . . . . Both componentsof J have a z dependence described with${k_{m}z} + \frac{( {{2p} - m} )\pi}{2}$

[0145] where p=0, 1 and m is an integer series commencing at p. Theindex p=0 describes basis functions that have J_(z)=0 at z=±L/2 whereasthe index p=1 describes basis functions with a J_(φ)=0 at z=±L/2.

[0146] If the specified field is symmetric with respect to the z=0plane, then all coefficients of even m terms will equal zero and hence mcan be assigned m=1,3,5, . . . . However, all terms of m are required ifthe specified region is asymmetric with respect to the z=0 plane as inthe example coil design presented herein.

[0147] The next step in the method is to calculate the coefficients ofthe basis functions in equations (30) and (31) such that a specifiedhomogeneous field is generated. Because the time harmonic Green'sfunctions are of a complex variable, the coefficients of the currentdensity J are complex as well. If the specified field is that of acircularly polarized field, then the real and imaginary components wouldbe the same albeit spatially separated by 90°. So if J(φ,z) producesKa_(x) in the DSV, J(φ−π/2,z) would produce Ka_(y) where K is just aconstant. The real part of the current that would produce a circularlypolarized field is then:

J _(linear)=real{J(φ,z)}−imag{J(φ−π/2,z)}  (33)

[0148] and this is the same current that would produce a linearlypolarized field. Hence only B_(x)â_(x) is needed to be specified in theDSV to produce J(φ,z) and (33) can be used to obtain the final currentdensity with only a real component.

[0149] Transmitting RF coils must have a shield to prevent eddy currentsin external conductors and to provide a suitable RF ground. A shield maybe approximated by specifying B=0 on a cylinder with a radius slightlylarger than the shield radius. A current is approximated on the shieldby the same set of basis functions as those for the inner coil exceptthat the shield current now has both div-free and curl-free components(see below).

[0150] If (a^(C)) and (c^(C)) denote column vectors of the J_(φ) andJ_(z) coefficients of the coil current (see equation (30)), (a^(S)) and(c^(S)) denote column vectors of the J_(φ) and J_(z) coefficients of theshield current (see equation (31)), and [˜] denotes a matrix, then thefield in the DSV due to the coil currents is found in matrix form, as:

[A](a ^(C))+[B](c ^(C))=(B _(xt) ^(C))   (34)

[C](a ^(S))+[D](c ^(S))=(B _(xt) ^(S))   (35)

[0151] where the summation of the vectors (B_(xt) ^(C)) and (B_(xt)^(S)) results in the target field vector (B_(xt))within the DSV. Similarmatrix equations result when applying B_(x)=0, B_(y)=0 and B_(z)=0 asthe condition at specified points near the outside of the surface of theshield cylinder:

[K](a ^(C))+[L](c ^(C))+[M](a ^(S))+[N](c ^(S))=(0)   (36)

[0152] The matrix equations are usually not square because the number ofpoints where the magnetic field is specified does not usually equal thenumber of unknown coefficients. The rank of the matrices are usuallyless than the number of unknown coefficients as well, and so aregularization method must be used for a solution.

[0153] It should be noted that equations (34), (35), and (36) can bewritten in more compact form as Eq. (I) above.

[0154] Regularization

[0155] The regularization method chosen as a non-limiting example was tominimize some functional in terms of the current density and impose theextra conditions onto the matrix equations such as (34) to (36).Numerous functionals are available and it is convenient to choose afunctional such that the resulting current density is easier toimplement. The functional: $\begin{matrix}{\min \quad {\int_{S_{0}}^{\quad}{{J \cdot J}{s}}}} & (37)\end{matrix}$

[0156] results in a current density similar to one of minimum power andhence is a useful choice. When the current density equations (30) and(31) are substituted into the above equation (37) and differentiated tofind the coefficients for a minimum: $\begin{matrix}{{\int_{S_{0}}^{\quad}{\frac{\partial}{\partial a_{ijkl}}( {J \cdot J} ){s}}} = 0} & (38) \\{{\int_{S_{0}}^{\quad}{\frac{\partial}{\partial c_{ijkl}}( {J \cdot J} ){s}}} = 0} & (39)\end{matrix}$

[0157] the resulting matrix equation is just some constant multiplied bythe identity matrix. This is because the basis functions are orthogonal.For a current density with no divergence, the wire implementationdepends on the stream-function n·S, where $\begin{matrix}{S = {{- n}\quad {\sum\limits_{q = 0}^{1}{\sum\limits_{p = 0}^{1}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = {p + 1}}^{M}{d_{mnpq}{\cos ( {{n\quad \varphi} + \frac{q\quad \pi}{2}} )}{\sin ( {{k_{m}z} + \frac{( {{2p} - m} )\pi}{2}} )}}}}}}}} & (40)\end{matrix}$

[0158] where n is normal to the cylindrical surface $\begin{matrix}{d_{mnpq} = \frac{\rho_{0}( {{\rho_{0}k_{m}a_{mnpq}} + {nc}_{mnpq}} )}{n^{2} + {\rho_{0}^{2}k_{m}^{2}}}} & (41)\end{matrix}$

[0159] Hence minimizing the variation in S over the surface wouldminimize the variation in its contours that ultimately determine thewire positions: $\begin{matrix}{\min \quad {\int_{S_{0}}^{\quad}{( {\nabla{\cdot {\nabla( {n \cdot S} )}}} )^{2}{s}}}} & ( {41A} )\end{matrix}$

[0160] where ∇·∇(n·S) gives a quantitative measure of thestream-function's variation.

[0161] In this case, because the basis functions on the coil form a setwith zero divergence, the d_(mnpq) coefficients of S reduce to:$\begin{matrix}{d_{mnpq} = \frac{a_{mnpq}}{k_{m}}} & (42)\end{matrix}$

[0162] Applying the condition (42) results in a matrix with diagonalterms: $\begin{matrix}{\lbrack R\rbrack = {\pi \quad {{L( {\frac{n^{\prime \quad 2}}{k_{m}\rho_{0}^{2}} + k_{m}} )}^{2}\lbrack I\rbrack}}} & (43)\end{matrix}$

[0163] [R] being the regularization matrix, and [I] being the identitymatrix.

[0164] Matrix Solution

[0165] An iterative method was used as a non-limiting example tocalculate the coil coefficients and shield coefficients separately sothat the error in each could be adjusted more exactly by adjusting twoscalar penalty values λ₁ and λ₂.

[0166] Firstly, the initial coil currents were calculated using theexpression:

(a _(i) ^(C))=([A′] ^(T) [A′]+λ ₁ [R ₁])⁻¹ [A′] ^(T)(B _(xt))   (44)

[0167] where [A′] is the combination of [A] and [B] of (34) using therelationship of equation (42) and [R₁] is the regularization matrixcalculated by applying (37) to the coil currents.

[0168] The field very close to the shield due solely to the coilcurrents is:

(b _(S/C))=[K′](a _(i) ^(C))   (45)

[0169] where [K′] is the combination of [K] and [L] of (36). Shieldcurrents are calculated to negate this field:

(s)=−([M′] ^(T) [M′]+λ ₂ [R ₂])⁻¹ [M′] ^(T)(b _(S/C))   (46)

[0170] where (s)^(T)=((a^(S))^(T) (c^(S))^(T)), [M′]=[[M] [N]] of (36)and [R₂] is the regularization matrix calculated by applying (37) to theshield currents. The field in the DSV due to the shield currents is:

(b _(DSV/S))=[C′](s)   (47)

[0171] where [C′]=[[C] [D]] of (35). The field in the DSV due to thecoil currents is:

(b _(DSV/C))=[A′](a _(i) ^(C))   (48)

[0172] The error in the DSV is then:

(b _(ε))=(B _(xt))−(b _(DSV/C))−(b _(DSV/C))   (49)

[0173] The new coil coefficient estimate is:

(a _(i+1) ^(C))=(a _(i) ^(C))+([A′] ^(T) [A′]+λ ₁ [R ₁])⁻¹ [A′]^(T)(b_(c))   (50)

[0174] The calculation then loops back to (45) and is terminated whenthe difference in the error between two consecutive iterations fallsbelow a certain predefined limit (e.g., 0.01). The error ε_(DSV) here isdefined as: $\begin{matrix}{ɛ_{DSV} = \frac{( b_{ɛ} )}{( b_{xt} )}} & (51)\end{matrix}$

[0175] and the shield error ε_(S) is defined as the error in the fieldproduced outside the shield:

ε_(S) =|b _(S/C) +[M′](s)|  (52)

[0176] The penalty scalars λ₁ and λ₂ affect the current density solutionas well as determining the condition of the matrices to be inverted inequations (44), (46) and (50). Typically, the matrix problem becomesunstable (that is, where the matrix is tending towards being singular)below a certain value of λ.

EXAMPLES ILLUSTRATING THE ASPECTS OF THE INVENTION IN WHICH CURRENTDENSITIES ARE CONVERTED TO DISCRETE COIL STRUCTURES

[0177] By whatever method the complex current density functions areobtained, the next step in the process of the overall design method isto synthesize the coil structure form the current density.

[0178] In a preferred embodiment, a stream function method is employedto calculate the positions and dimensions of conductors such that thestream function of the current distribution in the conductors closelyapproximates the stream function of the calculated current density.

[0179] Quasi-Static Case

[0180] It can be seen from equations (2) and (3) that the divergence ofthe quasi-static current density is zero:

∇·J=0   (53)

[0181] In true electromagnetic systems, the divergence is not zero but:

∇·J=−jωρ  (54)

[0182] where ρ is the charge density. However, for this analysis, thecharge distribution is assumed to be negligible, which is true forstatic and low frequency systems.

[0183] Because of zero divergence, a function S can be calculated thatrepresents the magnetic flux produced from the current density as:

∇×S=J   (55)

[0184] where from equations (2) and (3): $\begin{matrix}{S = {{- n}\quad \cos \quad \varphi {\sum\limits_{m = 1}^{M}{\frac{c_{m}}{k_{m}}\cos \quad k_{m}z}}}} & (56)\end{matrix}$

[0185] where n is the radial unit vector and is normal to thecylindrical surface. Note that if J is a surface current density, S willonly have a component perpendicular to the surface containing J, andhence only the magnitude of S is needed. The magnitude of S is afunction in two dimensions.

[0186] The current distribution in the conductors should be such that itproduces the same function S as that produced by the calculated currentdensity J. This is done by finding the contours of n·S and generatingappropriate currents in each of the wires placed in the positions of thecontours. If the contour ‘heights’ are chosen at equal incrementsbetween minimum and maximum, the currents in each of the wires followingthe contours should be equal. If the contour ‘heights’ are not equallyspaced, the currents in the wires have to generated accordingly inproportion to the contour spacing.

[0187] By way of an example, consider equation (56) with six realcoefficients (M=6): c_(m)=47.3518, 212.791, 314.615, 687.005, −3946.81and 5365.36.

[0188] The values of the stream function dictates the shape and positionof the contours and hence, is decided iteratively by investigating theelectromagnetic simulations each contour configuration produces. Forexample, the contours may be taken at equal fractions of thestream-function's maximum value, resulting in equal current in all thewires. Alternatively, if the maximum is normalized to one, contourscould be taken at ±0.697, ±0.394 and ±0.091. Because the interval(0.091) between zero and n·S=0.091 does not equal the interval (0.303)between n·S=0.091 and n·S=0.394, the current in the lowest 0.091 contourwill have to be 0.3 (i.e., 0.091/0.303) of the current flowing in the0.394 contour and 0.697 contour (these two contours should have equalcurrent).

[0189] The next task is to derive a workable model from these contours.Design decisions in this process include:

[0190] 1. A magnetic symmetry plane at φ=0 is used for the streamfunction in the range −L/2<z<L/2 and −π<φ<π.

[0191] 2. Each contour is implemented as a conductor. Two cases wereexamined: one where the conductors are implemented as circular wires(tubes) of specified diameter and the other where the conductors areimplemented as metal strips of specified width. The diameter or width ofa conductor determines its self-inductance and hence its compleximpedance. It is possible to adjust the current levels in the conductorsby adjusting the diameter or width of the conductors accordingly.

[0192] 3. The contours are connected depending on the sign of the streamfunction. If two contours are in close proximity and the stream functionvalue has the same sign for both contours, then the contours areconnected in parallel. If the contours have a stream function value ofdiffering sign, the contours are connected in series.

[0193] 4. Capacitors are added at the connections of the contoursconnected in series and also at the mid-point of contours that are notconnected to other conductors. However, there is a freedom to addcapacitors wherever the designer considers suitable. The values of thecapacitors are determined from an iterative approach using repeatedfrequency sweeps in the device simulation.

[0194] 5. The source points are at the extreme ends of the contour, thatis, either at both ends or at one end: either at z=L/2 and z=−L/2 oronly at z=L/2.

[0195] A full implementation of this method results in the conductorconfiguration shown in FIG. 7 for the current density defined by the sixreal coefficients set forth above, i.e., c_(m)=47.3518, 212.791,314.615, 687.005, −3946.81 and 5365.36.

[0196] In a preferred embodiment, asymmetric structures are synthesized.

[0197] When triangular basis functions are employed to approximate thecurrent density, only the current-density values are provided at pointswithin the domain of each triangular function. Thus the problem is tofind the appropriate contours of a two-dimensional grid-like data setthat will best approximate the current density that the data set ismeant to represent. In the quasi-static problem, the current density haszero divergence because of the nature of the basis functions and thusthe electric field at the surface is parallel with the current density.In the non-quasi-static case however, the electric field can have acomponent perpendicular to the current density and so an approach has toinclude measures for the approximation of the charge density. Inaccordance with the preferred embodiments of the invention, thecurl-free and divergence-free components of the current density arefound separately and each is approximated in turn.

[0198] Current-Density Components

[0199] The current density J can be split into its curl-free anddivergence-free components by writing:

J=∇×S+∇ψ  (57)

[0200] From Laplace's law, it follows that

∇² ψ=∇·J   (58)

[0201] and the curl-free component of J is:

J_(curl-free)=∇ψ  (59)

[0202] The divergence-free component can be found by applying curl toboth sides of equation (57) to give:

∇×∇×S=∇×J

[0203] If the divergence of S is made zero:

∇·S=0

then:

−∇² S=∇×J   (60)

[0204] The divergence-free component is:

J _(div-free) =∇×S   (61)

[0205] Since J_(div-free) is only on the surface, and to be free ofsingularities in the differentiation normal to a surface, it is best if

n×S=0

[0206] Hence, (60) also becomes a Poisson equation over the surface ofJ:

−∇²(n×S)=n·∇×J   (62)

[0207] The task then is to solve the two Poisson equations (58) and (62)for ψ and n·S and thereby using the contours of ψ and the contours ofn·S to approximate the entire current density J.

[0208] Note that J_(div-free) is the current density that directlyinfluences the magnetic field tangential to the surface of theconductor. J_(curl-free) is a measure of the quantity of rate of changeof charge density and this quantity directly influences the electricfield normal to the surface. In other words, J_(div-free) can be thoughtof as the inductive component of the circuit and J_(curl-free) as thecapacitive component of the circuit.

[0209] Once S and ψ are calculated, the contours and configuration ofthe coils can be designed. The contours of n·S dictate the currentconfiguration. Once n·S is found, its contours and hence the coilpattern can be calculated in a similar manner to the quasi-static case.

[0210] As examples of the above procedures, FIGS. 9 and 10 show theresults of determining inductive and capacitive elements for complexcurrent density functions obtained using the first method aspect of theinvention and shown in FIG. 8. The coil had a homogeneous region 50 mmfrom the center towards the lower end; the length of the coil was 320 mmand its radius was 100 mm. The operating frequency was 190 MHz.

[0211] A contour diagram of n·S is plotted in FIG. 9 and demonstratesthe paths for the conducting strips and/or wires. The contours of ψ willbe demarcation curves of varying areas of capacitance.

[0212] One mechanism to increase capacitance is to make an RF shield acontoured surface rather than just a cylindrical sheet, such that atregions of higher capacitance, the shield is closer to the coil than atregions where the capacitance is lower. Another mechanism is to adddielectric material of high permittivity at regions of highercapacitance compared to other regions, varying according to ψ.

[0213]FIG. 10 shows a contour plot of ψ for the same current density asshown in FIG. 8. The contours in this case are not current paths but aremarkers to indicate the regions of increasing charge density and hencecapacitance. The region of highest capacitive effect is in the center ofthe contours. Note that J_(curl-free) is actually perpendicular to thesecontour lines.

[0214] As further examples of the above procedures, FIGS. 11 through 15show the results of determining inductive and capacitive elements forcomplex current density functions obtained using the second methodaspect of the invention.

[0215] The design objective was to produce an RF coil 20 cm in diameter,25 cm in length with a DSV diameter of 10 cm, offset by 2.5 cm from thez=0 plane. (The z=0 plane passes half-way along the length of the coil).The coil was designed at the frequency of 190 MHz such that it could betested in an available MRI machine.

[0216] The second method of the invention was used with the constraintthat J_(z)=0 at z=±L/2. This constraint is a factor on how the resultingcoil is excited because no current distribution flows out from the edgeof the cylinder. After obtaining coefficients of the current density,the function S was calculated. The contours of the stream function n·Sfor half the coil are shown in FIG. 11. These are the preliminarypatterns for conductor positions. The resulting current density wastested using a commercial method of moments package (e.g., the FEKOprogram distributed by EM Software & Systems, South Africa). The coilcurrent density is approximated by Hertzian dipoles while the shieldcurrent density is ignored. Instead, a metallic shield approximated bytriangles as per the method of moments is positioned where the shieldshould be. The normalized magnetic field of the coil in the transverse(x,y) plane is shown in FIG. 12. The field varies within 10% over adistance of 13 cm in the x-direction (upper curve) and 12 cm in they-direction (lower curve). The field variation along the z-axis is shownin FIG. 13 which shows a 10% variation over a distance of 11 cm shiftedalong the z-axis by 2.5 cm. These simulated results generated byHertzian dipoles approximating coil currents and a metal cylinder forthe shield agree with the original field specification and target volumespecification in the inverse program.

[0217] The coil patterns were then converted to conductor patterns. Forthis case, 2.2 mm diameter wires were used for the conducting paths.Lumped elements were added to the model and adjusted such that thecurrent distribution at the resonant frequency of 190 MHz approximatedthe original calculated current distribution. The resulting coil isshown in FIG. 15 with the circuit diagram shown in FIG. 14. The valuesfor the capacitors are: C1=3.9 pF, C2=4.7 pF, C3=3.3 pF, C4=2.7 pF,C5=2.2 pF, C6=10 pF and C7 and C8 are variable capacitors.

[0218] When the coil was constructed, it realized an unloaded Q of 139,measured from the 3 dB down power points either side of the resonantfrequency In all of the examples presented throughout thisspecification, a homogenous target field has been specified, butnon-homogeneous target fields may be required in some circumstances andthese are readily handled by the methodology disclosed herein.

[0219] The invention has been described with particular reference toasymmetric radio frequency coils and a method for designing such coils,in which the section of interest (the DSV) can be placed at an arbitrarylocation within the coil. It will also be understood by those skilled inthe art that various changes in form and detail may be made withoutdeparting from the spirit and scope of this invention.

What is claimed is:
 1. A method for designing apparatus for use in amagnetic resonance system for receiving a magnetic resonance signalhaving a predetermined radio frequency, said apparatus and said magneticresonance system having a common longitudinal axis, said methodcomprising designing the apparatus by treating the apparatus as atransmitter of a radio frequency field having the predetermined radiofrequency and then designing said transmitter by: (a) defining a targetregion in which the radial magnetic component of the radio frequencyfield is to have desired values, said target region surrounding saidlongitudinal axis; (b) specifying desired values for said radialmagnetic component of the radio frequency field at a preselected set ofpoints within the target region; (c) determining a complex currentdensity function J, having real and imaginary parts, on a specifiedcylindrical surface by: (i) defining the complex current densityfunction as a sum of a series of basis functions multiplied by complexamplitude coefficients having real and imaginary parts; and (ii)determining values for the complex amplitude coefficients using aniterative minimization technique applied to a residue vector obtained bytaking the difference between calculated field values obtained using thecomplex amplitude coefficients at the preselected points and the desiredvalues at those points; and (d) converting said complex current densityfunction J into a set of capacitive elements located on the specifiedcylindrical surface and a set of inductive elements located on thespecified cylindrical surface by: (i) converting the complex currentdensity function into a curl-free component J_(curl-free) and adivergence-free component J_(div-free) using the relationships:J_(curl-free)=∇ψ, and J _(div-free) =∇×S, where ψ and S are functionsobtained from the complex current density function through theequations: ∇² ψ=∇·J, −∇² S=∇×J, and −∇²(n·S)=n·∇×J, where n is a vectornormal to the specified cylindrical surface; (ii) calculating locationson the specified cylindrical surface for the set of capacitive elementsby contouring the function ψ; and (iii) calculating locations on thespecified cylindrical surface for the set of inductive elements bycontouring the function n·S.
 2. The method of claim 1 wherein the basisfunctions are triangular and pulse functions.
 3. The method of claim 1wherein the iterative minimization technique is selected from the groupconsisting of linear steepest descent and conjugate gradient descent. 4.The method of claim 1 wherein (i) the set of inductive elements on thespecified cylindrical surface defined first and second ends for theapparatus and (ii) the target region has a midpoint that is closer tothe first end than to the second end.
 5. The method of claim 1comprising the additional step of displaying the locations on thespecified cylindrical surface for the set of inductive elements.
 6. Themethod of claim 1 comprising the additional step of producing the set ofinductive elements and the set of capacitive elements on the specifiedcylindrical surface.
 7. A method for designing apparatus for use in amagnetic resonance system for transmitting a radio frequency field orboth transmitting a radio frequency field and receiving a magneticresonance signal, said apparatus and said magnetic resonance systemhaving a common longitudinal axis, said method comprising: (a) defininga target region in which the radial magnetic component of the radiofrequency field is to have desired values, said target regionsurrounding said longitudinal axis; (b) specifying desired values forsaid radial magnetic component of the radio frequency field at apreselected set of points within the target region; (c) defining atarget surface external to the apparatus on which the magnetic componentof the radio frequency field is to have a desired value of zero at apreselected set of points on said target surface; (d) determining afirst complex current density function, having real and imaginary parts,on a first specified cylindrical surface and a second complex currentdensity, having real and imaginary parts, on a second specifiedcylindrical surface, the radius of the second specified cylindricalsurface being greater than the radius of the first specified cylindricalsurface by: (i) defining each of the complex current density functionsas a sum of a series of basis functions multiplied by complex amplitudecoefficients having real and imaginary parts; and (ii) determiningvalues for the complex amplitude coefficients using an iterativeminimization technique applied to a first residue vector obtained bytaking the difference between calculated field values obtained using thecomplex amplitude coefficients at the set of preselected points in thetarget region and the desired values at those points and a secondresidue vector equal to calculated field values obtained using thecomplex amplitude coefficients at the preselected set of points on thetarget surface; and (e) converting said first and second complex currentdensity functions into sets of capacitive elements and sets of inductiveelements located on the specified cylindrical surfaces by: (i)converting each of the first and second complex current densityfunctions into a curl-free component J_(curl-free) and a divergence-freecomponent J_(div-free) using the relationships: J_(curl-free)=∇ψ, and J_(div-free) =∇×S, where ψ and S are functions obtained from therespective first and second complex current density functions throughthe equations: ∇² ψ=∇·J, −∇² S=∇×J, and −∇²(n·S)=n·∇×J, where n is avector normal to the respective first and second specified cylindricalsurfaces and J is the respective first and second complex currentdensity functions; (ii) calculating locations on the respective firstand second cylindrical surfaces for the respective sets of capacitiveelements by contouring the respective functions ψ; and (iii) calculatinglocations on the respective first and second cylindrical surfaces forthe respective sets of inductive elements by contouring the respectivefunctions n·S.
 8. The method of claim 7 wherein the basis functions aretriangular and pulse functions.
 9. The method of claim 7 wherein theiterative minimization technique is selected from the group consistingof linear steepest descent and conjugate gradient descent.
 10. Themethod of claim 7 wherein (i) the set of inductive elements on the firstspecified cylindrical surface define first and second ends for theapparatus and (ii) the target region has a midpoint that is closer tothe first end than to the second end.
 11. The method of claim 7comprising the additional step of displaying the locations of the setsof inductive elements on the first and second specified cylindricalsurfaces.
 12. The method of claim 7 comprising the additional step ofproducing the sets of inductive and capacitive elements on the first andsecond specified cylindrical surfaces.
 13. A method for designingapparatus for use in a magnetic resonance system for receiving amagnetic resonance signal having a predetermined radio frequency, saidapparatus and said magnetic resonance system having a commonlongitudinal axis, said method comprising designing the apparatus bytreating the apparatus as a transmitter of a radio frequency fieldhaving the predetermined radio frequency and then designing saidtransmitter by: (a) defining a target region in which the radialmagnetic component of the radio frequency field is to have desiredvalues, said target region surrounding said longitudinal axis; (b)specifying desired values for said radial magnetic component of theradio frequency field at a preselected set of points within the targetregion; (c) determining a complex current density function J, havingreal and imaginary parts, on a specified cylindrical surface by: (i)defining the complex current density function as a sum of a series ofbasis functions multiplied by complex amplitude coefficients having realand imaginary parts; and (ii) determining values for the complexamplitude coefficients by solving a matrix equation of the form: [A](a^(C))=B where A is a transformation matrix between current density spaceand magnetic field space whose components are based on time harmonicGreen's functions, a^(C) is a vector of the unknown complex amplitudecoefficients, and B is a vector of the desired values for the magneticfield specified in step (b), said equation being solved by: (1)transforming the equation into a functional that can be solved using apreselected regularization technique, and (2) solving the functionalusing said regularization technique to obtain values for the complexamplitude coefficients; and (d) converting said complex current densityfunction into a set of capacitive elements located on the specifiedcylindrical surface and a set of inductive elements located on thespecified cylindrical surface.
 14. The method of claim 13 where theregularization functional is chosen so as to minimize the integral ofthe dot product of the complex current density function with itself overthe specified cylindrical surface.
 15. The method of claim 13 where thecomplex amplitude coefficients are chosen so that the complex currentdensity function has zero divergence.
 16. The method of claim 13 whereinstep (d) is performed by: (i) converting the complex current densityfunction into a curl-free component J_(curl-free) and a divergence-freecomponent J_(div-free) using the relationships: J_(curl-free)=∇ψ, and J_(div-free) =∇×S, where ψ and S are functions obtained from the complexcurrent density function through the equations: ∇² ψ=∇·J, −∇² S=∇×J, and−∇²(n·S)=n·∇×J, where n is a vector normal to the specified cylindricalsurface; (ii) calculating locations on the cylindrical surface for theset of capacitive elements by contouring the function ψ; and (iii)calculating locations on the cylindrical surface for the set ofinductive elements by contouring the function n·S.
 17. The method ofclaim 13 wherein the radio frequency field has a wavelength λ, theinductive elements on the cylindrical surface define a longitudinallength L, and L≧0.2λ.
 18. The method of claim 13 where the predeterminedradio frequency is at least 80 megahertz.
 19. The method of claim 13wherein (i) the set of inductive elements on the specified cylindricalsurface define first and second ends for the apparatus and (ii) thetarget region has a midpoint that is closer to the first end than to thesecond end.
 20. The method of claim 13 comprising the additional step ofdisplaying the locations for the set of inductive elements on thespecified cylindrical surface.
 21. The method of claim 13 comprising theadditional step of producing the set of inductive elements and the setof capacitive elements on the specified cylindrical surface.
 22. Amethod for designing apparatus for use in a magnetic resonance systemfor transmitting a radio frequency field or both transmitting a radiofrequency field and receiving a magnetic resonance signal, saidapparatus and said magnetic resonance system having a commonlongitudinal axis, said method comprising: (a) defining a target regionin which the radial magnetic component of the radio frequency field isto have desired values, said target region surrounding said longitudinalaxis; (b) specifying desired values for said radial magnetic componentof the radio frequency field at a preselected set of points within thetarget region; (c) defining a target surface external to the apparatuson which the magnetic component of the radio frequency field is to havea desired value of zero; (d) determining a first complex current densityfunction, having real and imaginary parts, on a first specifiedcylindrical surface and a second complex current density, having realand imaginary parts, on a second specified cylindrical surface, theradius of the second specified cylindrical surface being greater thanthe radius of the first specified cylindrical surface by: (i) definingeach of the complex current density functions as a sum of a series ofbasis functions multiplied by complex amplitude coefficients having realand imaginary parts; and (ii) determining values for the complexamplitude coefficients by simultaneously solving matrix equations of theform: [A ₁ ^(C)](a ^(C))+[A ₁ ^(S)](a ^(S))=B ^(C) [A ₂ ^(C)](a ^(C))+[A₂ ^(S)](a ^(S))=B ^(S) where A₁ ^(C), A₁ ^(S), A₂ ^(C), and A₂ ^(S) aretransformation matrices between current density space and magnetic fieldspace whose components are based on time harmonic Green's functions,a^(C) and a^(S) are vectors of the unknown complex amplitudecoefficients for the first and second complex current density functions,respectively, B^(C) is a vector of the desired values for the radialmagnetic field specified in step (b), and B^(S) is a vector whose valuesare zero, said equations being solved by: (1) transforming the equationsinto functionals that can be solved using a preselected regularizationtechnique, and (2) solving the functionals using said regularizationtechnique to obtain values for the complex amplitude coefficients; and(e) converting said first and second complex current density functionsinto sets of capacitive elements and sets of inductive elements locatedon the specified cylindrical surfaces.
 23. The method of claim 22 wherethe regularization functional is chosen to as to minimize the integralof the dot product of the first complex current density function withitself over the first specified cylindrical surface and to minimize theintegral of the dot product of the second complex current densityfunction with itself over the second specified cylindrical surface. 24.The method of claim 22 where the complex amplitude coefficients arechosen so that the first and second complex current density functionseach has zero divergence.
 25. The method of claim 22 wherein step (e) isperformed by: (i) converting each of the first and second complexcurrent density functions into a curl-free component J_(curl-free) and adivergence-free component J_(div-free) using the relationships:J_(curl-free)=∇ψ, and J _(div-free) =∇×S, where ψ and S are functionsobtained from the respective first and second complex current densityfunctions through the equations: ∇² ψ=∇·J, −∇² S=∇×J, and−∇²(n·S)=n·∇×J, where n is a vector normal to the respective first andsecond specified cylindrical surfaces and J is the respective first andsecond complex current density functions; (ii) calculating locations onthe respective first and second cylindrical surfaces for the respectivesets of capacitive elements by contouring the respective functions ψ;and (iii) calculating locations on the respective first and secondcylindrical surfaces for the respective sets of inductive elements bycontouring the respective functions n·S.
 26. The method of claim 22wherein the radio frequency field has a wavelength λ, the inductiveelements on the first cylindrical surface define a longitudinal lengthL₁, the inductive elements on the second cylindrical surface define alongitudinal length L₂, and L₁≧0.2λ, and L₂≧0.2λ.
 27. The method ofclaim 22 where the frequency of the radio frequency field is at least 80megahertz.
 28. The method of claim 22 wherein (i) the set of inductiveelements on the first specified cylindrical surface define first andsecond ends for the apparatus and (ii) the target region has a midpointthat is closer to the first end than to the second end.
 29. The methodof claim 22 comprising the additional step of displaying the locationsof the sets of inductive elements on the first and second specifiedcylindrical surfaces.
 30. The method of claim 22 comprising theadditional step of producing the sets of inductive and capacitiveelements on the first and second specified cylindrical surfaces.
 31. Amethod of converting a complex current density function J into sets ofcapacitive and inductive elements located on a specified cylindricalsurface comprising: (i) converting the complex current density functioninto a curl-free component J_(curl-free) and a divergence-free componentJ_(div-free) using the relationships: J_(curl-free)=∇ψ, and J_(div-free) =∇×S, where ψ and S are functions obtained from the complexcurrent density function through the equations: ∇² ψ=∇·J, −∇² S=∇×J, and−∇²(n·S)=n·∇×J, where n is a vector normal to the specified cylindricalsurface; (ii) calculating locations on the cylindrical surface for theset of capacitive elements by contouring the function ψ; and (iii)calculating locations on the cylindrical surface for the set ofinductive elements by contouring the function n·S.
 32. The method ofclaim 31 comprising the additional step of displaying the locations ofthe set of inductive elements on the specified cylindrical surface. 33.The method of claim 31 comprising the additional step of producing theset of inductive elements and the set of capacitive elements on thespecified cylindrical surface.